How can I prove <msubsup> <mi mathvariant="normal">&#x03A3;<!-- Σ --> <mrow class="MJX

Villaretq0 2022-06-27 Answered
How can I prove Σ i = 1 n 1 i = ( 2 n )
You can still ask an expert for help

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Lamont Adkins
Answered 2022-06-28 Author has 11 answers
Step 1
That i = 1 n i = n ( n + 1 ) 2 can be proved by induction.
The case k = 1 is trivial and assume it also holds for k = n 1. But then i = 1 n i = ( n 1 ) n 2 + n = n ( n 1 ) + 2 n 2 = n ( n + 1 ) 2 . Hence i = 1 n 1 i = n ( n 1 ) 2 .
Step 2
But by definition ( n 2 ) = n ! 2 ! ( n 2 ) ! = n ( n 1 ) ( n 2 ) ! 2 ( n 2 ) ! = n ( n 1 ) 2
Therefore we can conclude that i = 1 n 1 i = ( n 2 )

We have step-by-step solutions for your answer!

Jaqueline Kirby
Answered 2022-06-29 Author has 6 answers
Explanation:
Here’s a combinatorial proof: To choose two out of n elements (in ( n 2 ) ways), first choose the ( i + 1 )-th element and then choose one of the i elements preceding it.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2020-11-09
Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.
asked 2021-08-02
Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols:
asked 2021-07-28

Let A, B, and C be sets. Show that (AB)C=(AC)(BC)
image

asked 2021-08-18
Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b)R if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
asked 2021-08-22
Find the number of edges in a circulant graph Circ[n,{k,l}]. In mathematica you can explore these graphs by using the comand CirculantGraph [n,{k,l}.
asked 2022-05-14
(Solved)Find the number of 9 letter words using the letters P, Q, and R containing at least one P and at least two Qs.
Please help me with the last question on my discrete maths assignment because I can't get what I am doing wrong.
Find the number of 9 letter words using the letters P, Q, and R containing at least one P and at least two Qs.
number of 9 letter words that use all 3 letters = 3 9 .
A 1 = number of 9 letter words without the use of P = 2 9
A 2 = number of 9 letter words without the use of Q = 2 9
A 3 = number of 9 letter words using at least 1 Q = 3 9 2 9 .
A 1 A 2 = 1 9
A 1 A 3 = 2 9 1
A 2 A 3 = 0
A 1 A 2 A 3 = 0
number of words using at least 1 P and at least 2 Qs
= 3 9 ( 2 9 + 2 9 + ( 3 9 2 9 ) ( 1 + 2 9 1 0 ) + 0 ) = 0
I can't understand where I am making a mistake because everything seems to make sense.
asked 2022-04-16

 How many strings are there of lowercase letters of length four or less, not counting the empty string?

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question